document.write( "Question 1045312: Without using a calculator or technology, determine which is bigger, \r
\n" ); document.write( "\n" ); document.write( "\"sqrt%282017%29%5E%28sqrt%282016%29%29\" or \"sqrt%282016%29%5E%28sqrt%282017%29%29\".\r
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Algebra.Com's Answer #660729 by ikleyn(52810)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Without using a calculator or technology, determine which is bigger, \r
\n" ); document.write( "\n" ); document.write( "\"sqrt%282017%29%5E%28sqrt%282016%29%29\" or \"sqrt%282016%29%5E%28sqrt%282017%29%29\".
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document.write( "Let x = \"sqrt%282017%29%5E%28sqrt%282016%29%29\"  and  y = \"sqrt%282016%29%5E%28sqrt%282017%29%29\".\r\n" );
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document.write( "Then  \r\n" );
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document.write( "ln(x) = \"sqrt%282016%29%2Aln%28sqrt%282017%29%29\" = \"%281%2F2%29%2Asqrt%282016%29%2Aln%282017%29\"    and  \r\n" );
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document.write( "ln(y) = \"sqrt%282017%29%2Aln%28sqrt%282016%29%29\" = \"%281%2F2%29%2Asqrt%282017%29%2Aln%282016%29\".\r\n" );
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document.write( "Then\r\n" );
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document.write( "\"ln%28x%29%2Fln%28y%29\" = \"%28sqrt%282016%29%2Aln%282017%29%29%2F%28sqrt%282017%29%2Aln%282016%29%29\" = \"%28%28sqrt%282016%29%2Fln%282016%29%29%29%2F%28%28sqrt%282017%29%2Fln%282017%29%29%29\".\r\n" );
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document.write( "But  \"sqrt%28x%29%2Fln%28x%29\" is monotonically increasing function, as everybody knows (or everybody should know who studies/studied Calculus).\r\n" );
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document.write( "Therefore,  \"ln%28x%29%2Fln%28y%29\" = \"%28%28sqrt%282016%29%2Fln%282016%29%29%29%2F%28%28sqrt%282017%29%2Fln%282017%29%29%29\" < 1.\r\n" );
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document.write( "It implies that  ln(x) < ln(y).\r\n" );
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document.write( "Hence,  x = \"sqrt%282017%29%5E%28sqrt%282016%29%29\"  <  y = \"sqrt%282016%29%5E%28sqrt%282017%29%29\".\r\n" );
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document.write( "Answer.  \"sqrt%282017%29%5E%28sqrt%282016%29%29\"  <  \"sqrt%282016%29%5E%28sqrt%282017%29%29\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "By the way, there is a classic question: which number is bigger (or larger, or greater):\r
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\n" ); document.write( "\n" ); document.write( "\"e%5Epi\" or \"pi%5Ee\".\r
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\n" ); document.write( "\n" ); document.write( "The way to solve it is the same.\r
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\n" ); document.write( "\n" ); document.write( "For more elementary inequalities see the lesson\r
\n" ); document.write( "\n" ); document.write( "    - What number is greater? Comparing magnitude of irrational numbers \r
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\n" ); document.write( "\n" ); document.write( "(one more time) By the way, you have free of charge online textbook in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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